No one with CFS or ME would describe themselves as having chronic fatigue.
Do you mean as an illness description, or as a symptom description? Because as an illness description, yes, fully agreed, I think we're all aware that "chronic fatigue" as a diagnosis is like "chronic headache", just a description of an ongoing symptom, and not remotely the same as ME or CFS. But the actual symptom of chronic fatigue is a key part of having ME/CFS, so in that sense, yes, of course I would describe myself as having chronic fatigue, just as I would describe myself as having chronic pain and chronic GI problems. I've seen a few people saying things like, "ME is not about fatigue," or, "fatigue isn't the problem when you have ME," and I have always been utterly confused by them. Fatigue alone isn't the problem, absolutely, but it's the primary symptom, so how can you have ME without fatigue?
Weakness and exhaustion sound like fatigue to me. Do you mean that they are so much worse than what is usually meant by "fatigue" that it seems an inappropriate word to use, like saying "headache" for "migraine"? I think of it as more of a spectrum, I suppose.